Two characterizations are given of those structural consequence operations on a propositional language which can be defined via proofs from a finite number of polynomial rules.

This note concerns the ultraproduct construction. It is observed that in the class of SCI models ultraproducts may be constructed by a method apparently dierent from the standard one. Since the standard models of veryday model theory form a subclass of SCI models, one obtains a new view of ultraproducts. Aside from a few remarks, the paper is self-contained.

An abstract logic A, C consists of a finitary algebraA and a closure systemC onA. C induces two other closure systems onA, C P andC I, by projective and inductive generation respectively. The various relations amongC, C P andC I are determined. The special case thatC is the standard equational closure system on monadic terms is studied in detail. The behavior of Boolean logics with respect to projective and inductive generation is determined.

A signed -equation is an expression of the form t t or t t, where t and t are -terms (for some ranked set ). We characterize those classes of -algebras which are models of a set of signed -equations. Further we consider the problem of finding a complete deductive system analogous to equational logic for the logical consequence operation restricted to signed equations.